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A Node-based Smoothed Point Interpolation Method (NS-PIM) for Three-dimensional Thermoelastic ProblemsS. C. Wu ab; G. R. Liu bc; H. O. Zhang a; G. Y. Zhang c

a State Key Laboratory of Digital Manufacturing and Equipment Technology, Huazhong, University of Scienceand Technology, Wuhan, People's Repulic of China b Centre for Advanced Computations in EngineeringScience (ACES), Department of Mechanical Engineering, National University of Singapore, Singapore c

Singapore-MIT Alliance (SMA), Singapore

Online Publication Date: 01 January 2008

To cite this Article Wu, S. C., Liu, G. R., Zhang, H. O. and Zhang, G. Y.(2008)'A Node-based Smoothed Point Interpolation Method(NS-PIM) for Three-dimensional Thermoelastic Problems',Numerical Heat Transfer, Part A: Applications,54:12,1121 — 1147

To link to this Article: DOI: 10.1080/10407780802483516

URL: http://dx.doi.org/10.1080/10407780802483516

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A NODE-BASED SMOOTHED POINT INTERPOLATIONMETHOD (NS-PIM) FOR THREE-DIMENSIONALTHERMOELASTIC PROBLEMS

S. C. Wu1,2, G. R. Liu2,3, H. O. Zhang1, and G. Y. Zhang3

1State Key Laboratory of Digital Manufacturing and EquipmentTechnology, Huazhong, University of Science and Technology, Wuhan,People’s Repulic of China2Centre for Advanced Computations in Engineering Science (ACES),Department of Mechanical Engineering, National University of Singapore,Singapore3Singapore-MIT Alliance (SMA), Singapore

A node-based smoothed point interpolation method (NS-PIM) is formulated to analyze

3-D steady-state thermoelastic problems subjected to complicated thermal and mechanical

loads. Gradient smoothing technique with node-based smoothing domains is utilized to mod-

ify the gradient fields and to perform the numerical integration required in the weak form

formulation. Numerical results show that NS-PIM can achieve more accurate solutions

even when the 4-node tetrahedral mesh is used compared to the finite-element method

(FEM) using the same mesh, especially for strains and hence stresses. Most importantly,

it can produce an upper bound solution of the exact solution in energy norm for both

temperature and stress fields when a reasonably fine mesh is used. Together with FEM,

we now for the first time have a simple means to obtain both upper and lower bounds of

the exact solution to complex thermoelastic problems.

1. INTRODUCTION

Recently, meshfree methods [1] have been developed as the powerfulalternative techniques to the finite-element method (FEM) [2–4], and remarkableprogress has been made in providing more accurate and even upper bound solutionsto the exact ones in energy norms [5]. Widely used meshfree methods include thesmoothed particle hydrodynamic method (SPH) [6, 7], the element-free Galerkinmethod (EFG) [8], the reproducing kernel particle method (RKPM) [9], the meshlesslocal Petrov-Galerkin method (MLPG) [10], and the point interpolation method(PIM) [1], etc.

Received 29 July 2008; accepted 3 September 2008.

The support of the National Natural Science Foundation of China under project No. 50474053,

50475134, and 50675081 along with the 863 project (No. 2007AA04Z142) are gratefully acknowledged.

The authors also give sincere thanks to the support of the Centre for ACES, the Singapore-MIT Alliance

(SMA), and the National University of Singapore.

Address correspondence to S. C. Wu, State Key Laboratory of Digital Manufacturing and Equip-

ment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science

and Technology, Wuhan 430074, PR China. E-mail: [email protected]

1121

Numerical Heat Transfer, Part A, 54: 1121–1147, 2008

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407780802483516

Downloaded By: [Swets Content Distribution] At: 08:00 3 March 2009

The node-based smoothed PIM (NS-PIM or LC-PIM termed originally [11]) isfound very stable (spatially), works well with triangular types of mesh, is free fromvolumetric locking, and capable of producing upper bound solutions [5] in a veryconvenient manner using the same mesh as that used in standard FEM. The NS-PIMwas developed using shape functions constructed using simple point interpolationthrough a set of arbitrarily scattered nodes within a local support domain [12]. Bothpolynomial PIM [1] and radial PIM (RPIM) [13] of Delta function property can beused in the NS-PIM, which allows the straightforward imposition of essentialboundary conditions. NS-PIM or (NS-RPIM [14]) was originally formulated for2-D [11] and 3-D [15] elasticity problems with a more accurate stress solution andis more tolerant to mesh distortion, using the generalized gradient smoothing tech-nique [16] that was extended from the gradient smoothing technique [17] with nodalintegration scheme [18]. Therefore, NS-PIM allows incompatible displacement fields[16, 26]. The NS-PIM was then developed for 3-D heat transfer [19] and 2-D thermo-elasticity problems [20].

The NS-PIM or NS-RPIM can ensure at least linear consistency and hence theconvergence to the exact solution, and possesses a very important property of upperbound solutions for elasticity problems [5, 21]. The theoretical aspects of upperbound can be found in ref. [5]. Such properties for solution bounds in both tempera-ture and equivalent energy norms have also been observed numerically through anumber of heat transfers [19] and thermoelastic problems [20]. The NS-PIM worksvery efficiently with triangular and tetrahedral meshes that can be generated auto-matically for easily adaptive analysis of complicated geometries [22, 23]. The NS-PIM has also been proven to give a much higher convergence rate of up to 1.5 inenergy norm (observed as superconvergence [24]), in contrast to 1.0 which is thehighest possible rate for fully compatible linear FEM. The NS-PIM is also knownas a semi-equilibrium model for its properties of superconvergence in energy norm,immunity to volumetric locking, and satisfaction to equilibrium equations within allof the smoothing domains [5].

Based on the idea of NS-PIM, a node-based smoothed FEM (NS-FEM) [25]has also been formulated. The NS-FEM can be viewed as a special case of theNS-PIM and n-side polygonal element meshes can be used. It uses compatibledisplacement fields created based on element, and has quite similar properties as

NOMENCLATURE

a The vector of unkown coefficients

d nodal displacement vector

h convection coefficient, W=(m2 � �C)

ki thermal conductivity, W= (m � �C)

Qv internal heat source, W=m3

qC prescribed heat flux, W=m2

tC given traction, N=m2

TC specified temperature, �CTa known ambient temperature, �CuC given displacement, m

U vector of PIM shape functions

r; e; e0 cauchy stress and strain, initial strain�rr;�ee smoothed stress and strain

d;r variational and gradient symbol

a thermal expansion, 1=�CX integration or problem domain

C global or local boundary

Subscripts and superscripts

i, I, J tensor or node indices

k smoothing domain for node k

d thermal strain energy, J

T, T transpose symbol and energy indicator

1122 S. C. WU ET AL.


the NS-PIM. It has been found that both NS-PIM and NS-FEM cannot solvedynamic problems due to their ‘‘overly-softness’’ [5, 15, 19, 20] induced by the excess-ive node-based smoothing operations, which leads to temporal instability (spuriouseigenmodes) for solving time-dependent heat transfer and solid mechanics problems.

To tackle this problem, an edge-based smoothed FEM (ES-FEM) ES-PIM[40], and SFEM have recently been proposed for problems of solid mechanics [27],plate, and shells [28]. The ES-FEM can produce much more accurate solutions inboth the primary variable and its gradient using constant strain triangular and tetra-hedral element meshes. It is both partially and temporally stable, no spurious eigen-modes, and hence works well for dynamic problems.

With the increasing interests in optimizing physical processes of rapid heatingand solidification, analysis of thermal behaviors including temperature gradients aswell as resulted thermal stress becomes more and more important [29, 30]. Experi-mental study on these kinds of systems is usually very expensive, time-consuming,and practically difficult to obtain detailed results. Numerical means such as theFEM is often preferred for this kind of purpose [31]. However, due to its ‘‘overly-stiff’’ property, significant errors can occur especially in high gradient regions. The‘‘overly-stiff’’ nature of a full-compatible FEM model results in a lower bound sol-ution to the exact one. In contrast, the ‘‘soft’’ nature of the NS-PIM offers a veryuseful complementary property of upper bound solution. Thus, the simple combi-nation of the upper bound property of NS-PIM and the lower bound property ofFEM [4] can bound the numerical solutions from both sides for complicated realisticproblems, as long as a reasonablyfine mesh can be created.

In solving 2-D and 3-D problems with complicated geometries, meshing hasalways been an important issue [1]. It is the opinion of the authors’ group that theultimate solution to these systems is to use triangular and tetrahedral types of meshes,respectively, for 2-D and 3-D problems. Many meshfree methods [1, 12] and element-based methods [25, 27] have been formulated based onthis consideration.

In this work, the NS-PIM is further extended to 3-D thermoelastic problemsinvolving very complex geometries and complicated boundary conditions, includingboth thermal and mechanical loads. In they present NS-PIM procedure, we use thefour-node tetrahedral elements=cells for 3-D solids analysis that can be generatedwith ease. PIM shape functions are constructed using a polynomial basis and localsupporting nodes are based on these tetrahedral cells. Discretized system equationsfor problems of heat transfer and thermoelasticity are formulated using the general-ized smoothed Galerkin weak form [16]. The accuracy, convergence, and the mostimportant upper bound property of numerical solutions using the NS-PIM are stud-ied in detail in comparison with those obtained using the FEM.

2. PIM SHAPE FUNCTIONS

The polynomial PIM is a finite series representation method for creating shapefunctions, using a small set of nodes within a local support domain. In the scheme,the problem domain is first discretized with arbitrarily scattered nodes inside theproblem domain and on the boundary. The background tetrahedral cells are thenformed based on these nodes, as shown in Figure 1.

MESHFREE NS-PIM FOR 3-D THERMOELASTICITY 1123


Consider a continuous function u(x) for our 3-D thermoelastic problem, whichis defined in the problem domain X bounded by C,

uhðxÞ ¼Xn

i¼1

piðxÞai ¼ pTðxÞa ð1Þ

where pi(x) is a given basis function in the Cartesian coordinate space xT¼ [x,y, z], nis the number of polynomial terms, ai is the coefficient corresponding to the ith basisand is yet to be determined, and the superscript T stands for transpose operator.

The polynomial basis p (x) in Eq. (1) is usually built utilizing the Pascal’striangles and a complete basis is generally preferred for the requirement of a possiblehigher order of consistency. For complete order 1, the polynomial basis function canbe written as,

pðxÞ ¼ 1 x y zf gT ð2Þ

In order to determine the coefficients ai, a 3-D sub-domain (often called localsupport domain) for the interested point at x is constructed based on the background4-node tetrahedral cells shown in Figure 1, containing a total of n nodes. The coeffi-cients ai in Eq. (1) can then be determined by enforcing u(x) to pass through all nnodes, which leads to a set of n equations with each for one node.

u1 ¼ a1 þ a2x1 þ a3y1 þ a4z1 þ � � �u2 ¼ a1 þ a2x2 þ a3y2 þ a4z2 þ � � �

..

.

un ¼ a1 þ a2xn þ a3yn þ a4zn þ � � �

9>>>=>>>;

ð3Þ

Figure 1. Background 4-node tetrahedral cells for creating PIM shape functions.



Eq. (3) can be written in the matrix form of

Us ¼ Pa ð4Þ

where Us is the nodal displacement vector in the support domain, a is the vector ofunknown coefficients, and P is the coefficient matrix that can be given,

P ¼

1 x1 y1 z1 � � �1 x2 y2 z2 � � �1 x3 y3 z3 � � �... ..

. ... ..

. . ..

1 xn yn zn � � �

2666664

3777775

n�n

ð5Þ

Note that P is a square matrix because the number of nodes used in the localsupport domain is the same as the number of basis functions. If P�1 exists [1, 12], aunique solution for a can then be obtained as

a ¼ P�1Us ð6Þ

Different from the moving least-square approximation [32], coefficients a hereare constants even if the point of interest at x changes in a local cell of interest wherea same set of n nodes are used in the interpolation. Substituting Eq. (6) into Eq. (1)yields

uhðxÞ ¼ pTðxÞP�1Us ¼Xn

i¼1

uiðxÞui ¼ UTðxÞUs ð7Þ

where UðxÞ is a matrix of the PIM shape functions that can be expressed as

UTðxÞ ¼ pTðxÞP�1 ¼ u1ðxÞu2ðxÞ � � �unðxÞf g ð8Þ

The derivatives of the PIM shape function can be obtained very easily due toits polynomial property, but they are not required in the present NS-PIM because itis in fact a W2 formulation [26]. In addition, shape functions created using the PIMprocedure possess the Kronecker Delta function property, which permits simpleimposition of essential boundary conditions just as what we do in the FEM.

Note that in this work we only use the linear interpolation. Higher orderinterpolation and even the radial point interpolation are possible [14, 26]. In sucha case, the assumed field function is no longer continuous, belongs to a G space,and hence a more general W2 formulation based on the G space theory needs tobe used to ensure the stability, uniqueness, and the convergence [26].

3. DISCRETIZED SYSTEM EQUATIONS

The 4-node tetrahedral mesh, such as the one shown in Figure 1, is oftenpreferred in engineering analysis and design for 3-D problems with complicatedgeometry, because this kind of mesh can be generated much more easily in automatic



manners, resulting in significant savings in manpower and hence the cost in theanalysis and design process. Additionally, adaptive analysis can be more easilyimplemented for solutions of desired accuracy [22, 23]. We therefore only use4-node tetrahedral mesh as background cells for node selection and the PIMshape functions construction. When only four nodes are used in the local interp-olation it leads to a set of linear shape functions that are the same as those of thelinear FEM. Higher order NS-PIM using 4-node tetrahedral background meshcan be found in ref. [26].

3.1. Governing Equations

Consider a linear static thermoelasticity problem [33, 34] defined in an aniso-tropic solid X bounded by C. The governing equations (in the absence of body force)can be given in X as,

kxq2T

qx2þ ky

q2T

qy2þ kz

q2T

qz2þQv ¼ 0 ð9Þ

rr ¼ 0 ð10Þ

where T is the unknown field variable of temperature, ki (i¼ x, y, z) is theconductivity of heat in the ith direction, Qv is an inner heat source, and r denotesthe unknown stress field in the solids.

Three types of possible thermal boundary conditions are given by

T ¼ TC on CT Specified temperature ð11Þ

qn ¼ qC on Cq Specified heat flux ð12Þ

qn ¼ hðT � TaÞ on Cc Convection ð13Þ

where TC is the prescribed temperature, qC is the specified heat flux, h is theconvection coefficient, and Ta is the known temperature of the ambient medium.

The solid can also be subjected to mechanical conditions in the form ofspecified displacements and tractions,

u ¼ uC on Cu Specified displacement ð14Þ

r � n ¼ tC on Ct Specified traction ð15Þ

where uC is the prescribed displacement on Cu; tC is the given traction on Ct, and n isthe vector of unit outward normal on C.



3.2. Smoothed Galerkin Weak Form

In the general formulation of NS-PIM, a generalized smoothed bilinear formis used to establish the discretized system equations [16] and weakened weak (W2)formulation that allows assumed functions residing in a G space (including Hspace) is used [26]. In this work, to obtain the smoothed Galerkin weak formfor heat transfer analysis, the (energy) functional associated with heat equilibriumEq. (9) and its corresponding boundary conditions of Eqs. (11)–(13) can be firstexpressed as

IðTÞ ¼Z

X

1

2

"k1

qT

qx

� �2

þk2qT

qy

� �2

þk3qT

qz

� �2

�2QvT

#dX

�Z

Cq

qCTdCþ 1

2

ZCc

hðT � TaÞ2dC ð16Þ

Here, we require the assumed function for T has squarely integrable firstderivatives and hence is in a Hilbert (H) space. The variational form of the thermalsystem can now be presented as

dIðTÞ ¼Z

XdðrTÞTkrTh i

dX�Z

XdTQvdXþ

ZCq

dTqCdC

þZ

Cc

hTdTdC�Z

Cc

hTadTdC ¼ 0 ð17Þ

where k and rT are defined as

k ¼kx 0 00 ky 00 0 kz

8<:

9=; and rT ¼ qT

qx

qT

qy

qT

qz

� �T

ð18Þ

The temperature gradient rT presented in Eq. (17) is now replaced by thesmoothed temperature gradient rT . The smoothed Galerkin weak form for heattransfer problems can then be obtained as follows.Z

XdðrTÞTkrTdX�

ZX

dTTQvdXþZ

Cq

dTTqCdCþZ

Cc

dTThTdC

�Z

Cc

dTThTadC ¼ 0 ð19Þ

Substituting Eq. (7) into Eq. (19), a set of discretized system equations can bebuilt in the following matrix form.

½Kþ Kc�fqg ¼ ffg ð20Þ



in which the superscript c denotes the convective matrix and

KIJ ¼Z

XC

T

I kCJdX ð21Þ

KcIJ ¼

ZC3

hUTI UJdC ð22Þ

fI ¼Z

XUT

I QvdX�Z

C2

UTI qCdCþ

ZC3

hTaUTI dC ð23Þ

CI ¼ �ggIx �ggIy �ggIz

� �T ð24Þ

where �ggIx; �ggIy, and �ggIz are the x, y, and z components of the smoothed temperaturegradient for node I, respectively.

After obtaining the nodal temperature from solving Eq. (20), the strain of ther-mal expansion induced by the variation in temperature becomes the initial strain e0,

e0 ¼ axDT ayDT azDT 0 0 0� �T ð25Þ

where ai (i¼ x, y, x) is the thermal expansion coefficient in the ith direction. Therelation between the thermal stress and strain can be expressed as

r ¼ Dð�ee� e0Þ ð26Þ

where �ee and �rr are vectors containing the smoothed strains and stresses, respectively,and D is the matrix of material constants that is in general anisotropic [35]. For iso-tropic materials, D contains only Young’s modulus E and Poisson’s ratio n.

D¼ E

2ð1þ nÞð1� 2nÞ

2ð1� nÞ 2n 2n 0 0 02n 2ð1� nÞ 2n 0 0 02n 2n 2ð1� nÞ 0 0 00 0 0 1� 2n 0 00 0 0 0 1� 2n 00 0 0 0 0 1� 2n

26666664

37777775ð27Þ

In the standard weak formulation, the potential (strain) energy of the thermo-elastic system can be given by

PðuÞ ¼ 1

2

ZX

eTDedX�Z

XeTDe0dX�

ZCt

uTtCdC ð28Þ

where the strains are the so-called compatible strains obtained using the (linear)kinematics relation of the strains and the displacements. The variational form can



be expressed as

dPðuÞ ¼Z

XðdeÞTDedX�

ZXðdeÞTDe0dX�

ZCt

ðduÞTtCdC¼ 0 ð29Þ

In our formulation, the compatible strain e in Eq. (29) is replaced by thesmoothed strain �ee that satisfies the generalized (smoothed) Galerkin weak form thatcan be derived from the Hellinger-Reissner’s two-field variational principle. Thesmoothed Galerkin weak form for our thermoelastic problems can be written as,

dPðuÞ ¼Z

Xðd�eeÞTD�eedX�

ZXðd�eeÞTDe0dX�

ZCt

ðduÞTtCdC¼ 0 ð30Þ

Substituting Eq. (7) into Eq. (30), the discretized system equations for thermo-elastic problems can be expressed in the following matrix form.

½Ku�fdg ¼ fFg ð31Þ

where

Ku

IJ ¼Z

XB

T

I DBJdX ð32Þ

FI ¼Z

Ct

UTI tCdCþ

ZX

BT

I De0dX ð33Þ

in which B is the smoothed strain matrix that is obtained through the gradientsmoothing operation, and the superscript u denotes the stiffness matrix associatedwith the displacement field of a given discretization. To obtain the smoothed stiff-ness matrix of thermoelastic system, the nodal integration scheme with node-basedgradient smoothing technique will be used to perform the domain integration, whichwill be detailed in the following section.

3.3. Gradient Smoothing Over Node-Based Smoothing Domains

To carry out the domain integration required in Eqs. (21) and (32), a set ofnode-based nonoverlapping sub-domains is created on top of the background meshof 4-node tetrahedrons used for shape function construction. This set ofsub-domains is also used for the gradient smoothing, and hence called smoothingdomains. Based on the tetrahedral mesh, N smoothing domains Xkðk ¼ 1; 2; . . . ;NÞcentered by node k can be formed. Figure 2 shows a typical nodal smoothingdomain for one node and the details on the formation is illustrated in Figure 3.The smoothing domain of the interested node k is constructed by connectingsequentially the mid-edge-points, the centroids of the surface triangles, and the cen-troids of the sample cell I. The boundary of the smoothing domain Xk is labeled asCk and the union of all Xk forms the global domain X exactly. Using this set ofnode-based smoothing domains, the domain integration in Eqs. (21) and (32)



becomes simple summation for all the smoothing domains, and the ‘‘stiffness’’matrices can be further expressed as

KIJ ¼XN

k¼1

KðkÞIJ ð34Þ

Figure 2. A typical 3-D node-based smoothing domain for node k formed by sequentially connecting the

mid-edge-points, centroids of surface triangles, and centroids of eight surrounding four-node tetrahedrons.

Figure 3. The schematic of one partition of the node-based smoothing domain for node k as one vertex of

four-node tetrahedral cell I (k-k2-k3-k6).



The sub-stiffness matrices for node k has the form of

KðkÞIJ ¼

ZXk

CT

I k CJ dX ð36Þ

�KKuðkÞIJ ¼

ZXk

BT

I D BJ dX ð37Þ

The gradient smoothing operation [17] is now performed over the smoothingdomains, which yields the following smoothed gradient.

�eehIJðxkÞ ¼

ZXk

ehIJðxÞ Wðx� xkÞ dX ð38Þ

where ehIJ is the compatible gradient and W is a smoothing function given by

Wðx� xkÞ ¼1=Vk x 2 Xk

0 x =2Xk

�ð39Þ

where Vk ¼R

XkdX is the volume of smoothing domain for node k, as shown in

Figure 2. Substituting Eq. (39) into Eq. (38) and integrating by parts, the smoothedgradient can become

�eehIJðxkÞ ¼

1

Vk

ZXk

ehIJðxÞ dX

¼ 1

Vk

ZXk

1

2

quhI

qxJþ quh

J

qxI

� �dX

¼ 1

2Vk

ZCk

ðuhI nJ þ uh

JnI ÞdC ð40Þ

where Ck is the boundary of the smoothing domain for node k.Using the PIM shape functions for the temperature and displacement interpo-

lations in the form of Eq. (7), the smoothed strain and temperature gradient for nodek can be further expressed in the following matrix form, respectively, as

�eehðxkÞ ¼XI2Dk

�BBIðxkÞUI ð41Þ

ghðxkÞ ¼XI2Dk

CIðxkÞqI ð42Þ

where Dk is the number of total nodes associated with the smoothing domain of nodek. For three-dimensional spaces, the corresponding expressions are

g ¼ f�ggx �ggy �ggzgT �ee ¼ f�eex �eey �eez �eexy �eeyz �eexzgT ð43Þ



qI ¼ fTI TI TIgT UI ¼ fUIx UIy UIzgT ð44Þ

CI ¼ �bbIx�bbIy

�bbIz

� �TB

T

I ðXkÞ ¼�bbIx 0 0 �bbIy 0 �bbIz

0 �bbIy 0 �bbIx�bbIz 0

0 0 �bbIz 0 �bbIy�bbIx

24

35 ð45Þ

and

�bbIi ¼1

Vk

ZCk

uI ðxÞniðxÞdC ði ¼ x; y; zÞ ð46Þ

where uI(x) is the PIM shape function for node I. It can be found that the volumeintegrals are now transformed into surface integrals involving only the shape func-tion due to the use of the gradient smoothing technique. Such a transform is exact,as long as the shape functions are continuous in the problem domain (in an H space).Otherwise, it is an approximation (in a G space) [16].

Using Gauss integration along boundary surface Ck of the smoothing domainXk, Eq. (46) can be finally written in the following summation.

�bbIi ¼1

Vk

XNs

q¼1

XNg

r¼1

wruIðxqrÞniðxqÞ" #

ð47Þ

where Ns is the number of segment surface on Ck, Ng is the number of total Gausspoints used for each segment surface, and wr is the weight corresponding to theGauss point.

Using the smoothed gradient and strain shown in Eq. (45), and furtherassuming that the gradient is constant over the entire smoothing domain, thestiffness matrices for our thermoelasticity problems can be obtained by simplesummation.

KIJ ¼XN

L¼1

CCTI ðxLÞ k CJðxLÞVL ð48Þ

Ku

IJ ¼X3N

L¼1

�BBTI ðxLÞD�BBJðxLÞVL ð49Þ

where VL is the volume of smoothing domain for given node L.It can be easily observed from Eqs. (48) and (49) that the resultant linear sys-

tem matrices are symmetric and banded (due to the compact supports of PIM shapefunctions), which implies that the discretized system equations can be solvedefficiently. Note, that there is no increase in degrees of freedom in the NS-PIMmodel, and hence, the dimensions of the matrices for NS-PIM are exactly the sameas those in the FEM so long the same mesh is used.



4. NUMERICAL EXAMPLES

Two mechanical components subjected to thermal and mechanical loads areanalyzed using the present NS-PIM and examined in detail in terms of the accuracy,convergence, and important bound property of numerical solutions. For compari-son, an in-house FEM code is also used to simulate the problem using the samemeshes as NS-PIM. As the analytical solutions are not available for these problems,reference results are obtained using ABAQUS1 with a very fine mesh for compari-son purposes. The equivalent energy norm heat transfer problem [4] is defined as

UT ¼Z

X�ggTk�gg dX ð50Þ

where, in our current case, �gg is the smoothed temperature gradient in Eq. (43).The temperature effect can be measured in the form of thermal strain

energy [36]

Ud ¼1

2

ZXð�ee� e0ÞTdX

¼ 1

2

ZXð�eeTD�ee� 2�eeTDe0 þ eT

0 De0ÞdX ð51Þ

where the first term in Eq. (51) is the smoothed stiffness matrix derived earlier, whichis a bilinear functional that represents the mechanical strain energy in the system dueto the thermal effect. The last term is a constant with respect to the assumed mech-anical displacements and hence has no effect on the stationary state of the system.The middle term is a linear functional, and is to be assembled to the system as aglobal force vector representing the temperature load.

In our analysis, the thermal analysis is performed first to obtain the distri-bution of the temperature in the solid, and then the thermoelastic analysis is conduc-ted for the distributions of displacement, strain, and von Mises stress as theconsequences of both the thermal effect and external mechanical loads.

4.1. 3-D Conduction Beam

The first example considered here is a 3-D conduction beam subjected to ther-mal loads, as shown in Figure 4. The temperature TC is prescribed onto the left sur-face, a heat flux qC enters into the solid continuously from the right surface, and heatconvection occurs between the bottom surface and the ambient with a convectioncoefficient h.

The beam is also loaded with mixed mechanical pressure on the top surface,which varies in the form of (�15þ 5000x2)Pa, as shown in Figure 5. The positivevalue in traction stands for the tensile and negative denotes the suctions. Note theessential boundary conditions of both temperature and displacement on the leftsurface can be imposed directly on the nodes thanks to the Delta function propertyof the PIM shape functions. The parameters used in the computation are:L¼ 0.09 m, H¼ 0.009 m, B¼ 0.009 m, Qv¼ 0, k1¼ 15 W=(m � �C), k2¼ 10 W=(m � �C),



k3¼ 5 W=(m � �C), h¼ 1500 W=(m2 � �C), TC¼ 0, Ta¼ 400�C, qC¼�2000 W=m2,uC¼ 0, ax¼ ay¼ az¼ 1.02� 10� 5=�C, E¼ 3.0� 1010 Pa, and n¼ 0.3.

The problem domain is discretized with 1,042 irregularly distributed nodes toform the 4-node tetrahedrons (T4). The same mesh is used for both the NS-PIMmodel and our in-house FEM model. The reference solution is obtained using a veryfine mesh with about 10 times more nodes.

4.1.1. Temperature Distribution. A heat transfer analysis is performed first,and the computed temperature values obtained using the present NS-PIM at thenodes on the bottom edge (marked as AB in Figure 4) are listed in Table 1, alongwith the results of linear FEM and the reference solution. It can be found thatNS-PIM solutions are in very good agreement with the reference solutions, and havea similar accuracy as the FEM solutions. This shows that the NS-PIM model workswell for this 3-D heat transfer problem. This finding is consistent with those made inprevious studies [15, 20]; NS-PIM gives at least a similar accuracy for the results ofprimary variables (that is temperature in this case).

Figure 4. Discretized model of a 3-D beam subjected to thermal and mechanical loads.

Figure 5. Mechanical loading curve: pressure on the top surface of the 3-D beam.



4.1.2. Thermal Deformation. After the temperature field is obtained, theinitial strain resulted from the (free) thermal expansion can be obtained based onEq. (25), and the temperature load is then added to the global load vector of the ther-moelastic system. The NS-PIM is then used again to compute the deformation of the3-D beam as a consequence of both the thermal and mechanical loads. Figures 6aand 6b show, respectively, the distribution of displacements in x and y directions,together with the reference solutions that are obtained using fine mesh with a totalof 11,052 nodes.

It can be observed that the computed displacements match well with the refer-ence, and the accuracy of linear NS-PIM is better than that from the linear FEMusing the same 4-node tetrahedral mesh. This shows the effectiveness of the NS-PIM formulation for this 3-D thermoelastic problem.

Figure 7 shows the contours of displacement in the z-direction obtained fromboth NS-PIM and FEM, together with the reference obtained using irregularly dis-tributed 14,873 nodes. It can be observed that the present results are in better agree-ment with the reference solution compared to those of the FEM when the same meshis used. Note also that the largest displacement is in even better agreement with thereference solution compared to the FEM’s solution.

4.1.3. Von Mises Stress. The von Mises criterion is the most commonly usedfor the assessment of stress failure of metal materials [33, 36]. The failure criterionstates that the material will fail when the von Mises stress rVM exceeds the allowable

Table 1. Comparison of the solutions of nodal temperature (�C) along the AB edge for the 3-D beam

problem

x (m) 0.003 0.012 0.021 0.030 0.039 0.048 0.057 0.660 0.075 0.084 0.090

Reference 228.84 357.56 388.73 396.99 399.23 399.81 399.96 400.03 400.19 400.73 401.92

NS-PIM 243.53 357.75 388.64 396.97 399.24 399.82 399.96 400.03 400.18 400.70 401.90

FEM 212.71 357.60 388.49 397.04 399.21 399.80 399.96 400.03 400.17 400.69 401.88

Figure 6. Distribution of the displacement along edge AB on the 3-D beam subjected to thermal and

mechanical loads.



yield stress rY of the given material. Hence we need to make sure at any point in thesolid=structure that

rVM � rY ð52Þ

The von Mises stress rVM is computed using

rVM �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI2

1 � 3I2

qð53Þ

Figure 7. Comparisons of computed displacement field in the z-direction.



where I1and I2 are the first two invariants of the stress tensor. Using the values ofstress components obtained numerically, I1and I2can be computed using

I1 ¼ �rrx þ �rry þ �rrz ð54Þ

I2 ¼ �rrx �rry þ �rry �rrz þ �rrx �rrz � �ss2xy � �ss2

yz � �ss2xz ð55Þ

Figure 8 plots the contour of von Mises stress obtained using the NS-PIM, thereference means, and FEM. Figure 8 reveals that the present NS-PIM can obtainmore accurate von Mises stress than FEM does, and the NS-PIM solution obtainedusing a very coarse mesh of 505 nodes agrees very well with the reference one.

Figure 8. Comparisons of contours for nodal von Mises stress (rVM ) in the 3-D beam.



Note that the present NS-PIM formulation is derived from the smoothedGalerkin weak form, in which the smoothed gradients (�ee and �gg) are obtained throughEqs. (41) and (42), respectively. The NS-PIM model so-constructed behaves softercompared with the FEM model [11], and hence can produce a more accurate andmore smoothing solution for the gradients (strains and stresses). The similarphenomenon has also been observed in the analysis of solid mechanics [11, 15].NS-PIM can obtain higher accuracy in stresses than the corresponding FEM modelbased on the same mesh.

4.1.4. Solution bound. It is well-known that the displacement-based fullycompatible FEM always provides a lower bound [4] in the energy form of the exactsolution to elastic problems with homogeneous essential boundary conditions. It is,however, much more difficult to bound the solution from above for thermoelasicproblems, and the NS-PIM offers one effective means for the upper bound solution.To present the important bound property of NS-PIM, four models for the 3-D beamare created with irregularly distributed nodes of 505, 1,042, 1,667, and 2,858.

Figure 9 shows the convergence property of the equivalent energy for the tem-perature field with the increase of degree of freedoms (DOF) for both NS-PIM andFEM. The reference value is calculated using ABAQUS1 with a very fine mesh of14,873 nodes. It can be seen that the equivalent energy obtained using the NS-PIMmodel is larger than that of the reference solution; in comparison, the energy of theFEM model is smaller than the reference value. This finding confirms that the 3-DNS-PIM formulation provides an upper bound solution for the heat transfer problem[20], which is an important complement to the fully compatible FEM.

Figure 10 shows the convergence property in strain energy for the thermoelas-tic system; NS-PIM gives an upper bound and FEM gives a lower bound. The

Figure 9. Upper and lower bound solutions in equivalent energy for the temperature field in the 3-D beam

obtained using the present NS-PIM and FEM.



thermal strain energy of the NS-PIM model is larger than the reference solution andconverges from above with the increase in DOF (or mesh density). On the contrary,the corresponding FEM model approaches the exact model from below and obtainsthe lower bound solution for thermoelasticity problems with homogeneous essentialboundary conditions in both temperature and displacement.

4.2. An Engine Pedestal

The following example comes from an actual aero-engine pedestal witha very complicated geometry, which is manufactured using the plasma

Figure 10. Upper and lower bound solutions in strain energy for 3-D thermoelastic beam obtained using

the present NS-PIM and FEM.

Figure 11. An engine pedestal subjected to Dirichlet, Neumann, and Robin conditions, respectively, on

the baseplane, inner surface of installing hole, and circumferential surface of the cap structure. Displace-

ment and traction boundary conditions are also applied on the baseplane and circumferential surface.



deposition-layered technique [31]. The pedestal part is made of superalloymaterial and the dimensions can be found in ref. [37]. A simplified model of ped-estal part is shown in Figure 11.

The thermal loadings are also quite complicated and are given in the form ofspecified temperature, heat flux, and convection on different situations of the ped-estal: the baseplane, inner surface of the installing hole, and the circumferential sur-face, as indicated by color codes in Figure 11. In addition, the pedestal is alsosubjected to mechanical loads in which the baseplane is fixed rigidly with zero dis-placement in three directions, and a constant traction of -5� 105Pa is given to thecircumferential surface to simulate the compressive effect of mechanical loads origi-nated from other components and whole mechanical structures.

For quantitative comparison, the circumferential surface CD of the baseplane(marked in Figure 11) is divided evenly into eight segments, with nine observationpoints. The coordinates of the nine observation points on the circumferential arc edgeCD are listed in Table 2. Computed values at these observation points will be outputand examined in great detail. Other parameters used in the computation of the ped-estal part are the same as those for the 3-D beam problem listed in Section 4.1.

4.2.1. Temperature distribution. The problem domain is discretized with atetrahedral mesh of 587 nodes. The computed temperature values at these nineobservation nodes are listed in Table 3, together with the FEM solution and the ref-erence solution obtained using 12,344 nodes. It is found that the NS-PIM results oftemperature are larger than the reference solutions, while the FEM solutions aresmaller than the reference solutions. This attributes to the softening effect of theNS-PIM model induced by the gradient smoothing technique. These findings showagain that together with FEM, the NS-PIM can bound the solution from both sidesusing the same elemental mesh.

4.2.2. Displacement field. Due to the combined action of thermal and mech-anical loads, the displacement field within the pedestal is expected to be more compli-cated. Figure 12 shows the distribution of the displacements in the x and y directionsalong the circumferential arc CD. It is clearly seen that the solutions obtained from

Table 2. Cartesian coordinates x (x, y, z) of the nine nodes located on the circumferential arc CD

ID. 1 2 3 4 5 6 7 8 9

x(m) 8.0E-3 6.8E-2 4.8E-2 2.5E-2 9.5E-3 1.4E-2 3.4E-2 5.7E-2 7.3E-2

y(m) 2.E-17 1.8E-2 3.2E-2 2.8E-2 9.8E-3 �1.8E-2 �3.2E-2 �2.8E-2 �4.9E-2

z(m) 4.1E-3 1.3E-2 8.8E-3 5.6E-3 4.2E-3 4.7E-3 6.5E-3 1.1E-2 1.4E-2

Table 3. Comparisons of numerical solutions of temperature (�C) along the circumferential arc CD

ID. 1 2 3 4 5 6 7 8 9

Reference 318.39 296.45 321.51 323.97 317.94 318.47 327.04 310.64 296.75

NS-PIM 320.09 300.92 323.67 325.67 319.52 320.00 330.13 313.47 300.65

FEM 316.41 290.58 317.99 319.90 315.54 314.42 324.39 304.50 293.42



Figure 12. Distribution of displacement along the circumferential arc CD on the pedestal subjected to

both thermal and mechanical loads.

Figure 13. Comparisons of computed thermal strain.



the NS-PIM agree well with the reference solutions (obtained using 11,636 nodes),and much better than FEM using the same 4-node tetrahedral mesh.

4.2.3. Strain and von Mises stress. Figure 13 shows the contours of strainfields obtained using NS-PIM, FEM of the same mesh, and the reference solution. Itis observed that the contour lines of NS-PIM solutions are smoother than those ofFEM. The NS-PIM results are also much closer to the reference results, comparedwith that of the FEM.

Figure 14 demonstrates the comparison of the resultant von Mises stress con-tour in the pedestal. It can be clearly seen that the NS-PIM solution, especially forthe nodal peak von Mises stress within high-gradient region, agrees better with thereference one compared with the FEM solution using the same coarse mesh.

4.2.4. Solution bound. Figures 9 and 10 show that the NS-PIM solution(in both equivalent energy for heat transfer and strain energy for thermoelasticity)is larger than the reference solution. The reference solution is, on the other hand,

Figure 14. Comparisons of computed von Mises stress.



larger than that of the displacement-based fully compatible FEM. This boundproperty of NS-PIM model is of practical importance in providing certified solu-tions for engineering problems, and hence is examined in detail in the 3-D beamproblem. To further confirm the upper bound property of the present NS-PIMfor more complicated thermoelastic systems, four discretized models of the pedestal

Figure 15. Upper and lower bound solutions in equivalent energy for the temperature field in the 3-D

pedestal.

Figure 16. Upper and lower bound solutions in strain energy for 3-D pedestal.



are created with irregularly distributed 587, 1,117, 1,543, and 2,315 nodes.Figure 15 plots the numerical solutions (in equivalent energy norm for the tem-perature field) against the DOFs for both NS-PIM and FEM. The reference sol-ution is obtained using the FEM with a very fine mesh of 12,344 nodes. It canbe observed again that the present NS-PIM produces an upper bound solutionin equivalent energy for this very complicated problem with homogenous essentialboundary conditions (TC¼ 0). The solution converges to the reference solutionfrom above with the increase of DOFs.

Figure 16 plots the convergence state of numerical solutions in thermal strainenergy for this complicated pedestal problem with homogenous essential boundaryconditions (uC¼ 0). It can be seen that the present NS-PIM also provides an upperbound solution in thermal strain energy. The upper bound property of the NS-PIMimplies that even for the primary variables (nodal temperature q in Eq. (20) andnodal displacement d in Eq. (31)) the solution can also be larger than that of theexact model under certain conditions. This has been observed in both Tables 1and 3, where the values are all positive (and hence has the same feature of an energynorm).

Figures 9, 10, 15, and 16 indicate that with the increase of DOF, bothequivalent energy for the temperature field and the strain energy for thermoelasti-city obtained from NS-PIM and FEM all converge to the referencesolution fromabove and below, respectively. This important property allows engineers to verifya numerical solution and conduct the adaptive analysis for solutions of the desiredaccuracy for complicated solids and structures subjected to complicated thermaland mechanical loads when the analytical solutions are difficult to obtain. Movingforward, one can even attempt to obtain exact solutions in certain norms usingonly finite mesh, by properly combining softer and stiffer models, as reported inrefs. [38, 39].

5. CONCLUSIONS

In this work, the NS-PIM is further formulated for 3-D thermoelastic problemswith complex geometric shape and complicated boundary conditions under the var-ied thermal and mechanical loads. In this approach, polynomial basis functions areutilized to construct the PIM shape functions using the point interpolation pro-cedure. The smoothed Galerkin weak form is then adopted to create the discretizedsystem equations. Examples of 3-D complex thermoelastic problems are analyzed toexamine the accuracy, convergence, and the very important upper bound property ofNS-PIM. Several concluding remarks can be made from our detailed study.

1. No derivatives of shape functions are required due to the use of the gradientsmoothing technique, which can result in more accurate gradient solutions evenusing low-order shape functions.

2. Shape functions generated using the point interpolation method possess the Kro-necker Delta function property, and hence allow the straightforward treatment ofessential boundary conditions.

3. The present NS-PIM can achieve higher accuracy in von Mises stress comparedwith FEM, using the same lower-order shape functions.



4. For the first time, the upper bound solutions in both equivalent energy norm forheat transfer and thermal strain energy norm for thermoelasticity are obtainedusing the NS-PIM for 3-D thermoelastic problems with homogeneous essentialboundary conditions. Together with the standard FEM, we now have a simplemeans to bound the solution for thermoelastic problems from both below andabove using the same tetrahedral mesh.

In order to analyze the time-dependent problems, an efficient edge-basedsmoothed point interpolation method (ES-PIM) [40] has been proposed recentlyfor solid mechanics. The meshfree ES-PIM is then further formulated for transientlinear and nonlinear heat transfer problems.

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